Glauert's optimum rotor disk revisited - a calculus of variations solution and exact integrals for thrust and bending moment coefficients

The present work is an amendment to Glauert's optimum rotor disk solution for the maximum power coefficient, CP_max , as a function of tip speed ratio, λ. First, an alternate mathematical approach is pursued towards the optimization problem by means of calculus of variations. Secondly, analytical solutions for thrust and bending moment coefficients, CT and C_Be, are derived, where an interesting characteristic is revealed pertaining to their asymptotic behavior for λ → ∞. In addition, the limit case of the non-rotating actuator disk for λ → 0 is shown for all three performance coefficients by repeated use of L'Hôpital's theorem, and its validity is discussed in the context of other works since Glauert.